Epoxy resins modified with only Polyhedral Oligomeric Silsesquioxanes (POSS) or cubic boron nitride (c-BN), as well as multiphase composites containing both, POSS and c-BN, were fabricated and investigated. Two different types of reactive liquid POSS with molecule sizes of approximately 2 nm, viz. Triglycidylisobutyl-POSS (TGIB-POSS) and Glycidyl-POSS were added in contents of 1 wt% to form single-phase and multiphase composites. Submicrometric c-BN, in contents of 5 wt%, was used for the respective single-phase composite as well as in the multiphase samples to create multifunctional nanostructured composites. All specimens were characterized by Broadband Dielectric Spectroscopy (BDS), dielectric breakdown experiments, Differential Scanning Calorimetry (DSC) and thermal conductivity measurements. The single-phase POSS composites have seen up to 15% improved dielectric breakdown strengths and enhanced thermal conductivities by up to 20%. The single-phase c-BN composite revealed the most distinct improvement in thermal conductivity of more than 25%, along with a reduction in dielectric breakdown strength of approximately 18%. Additionally, the dielectric and thermal properties of the multiphase composites containing 1 wt% POSS and 5 wt% c-BN, were found to be solely dictated by the submicrometric c-BN particles. SEM observations revealed a significantly improved dispersion of the c-BN particles in the presence of POSS for the multiphase composites, suggesting that POSS acts as a compatibilizer between organic epoxy matrix and inorganic filler particles.